Quote:
Originally Posted by gd_barnes
Another interesting effort would be to check k*b^n1 primes for k*b^n3 primes to make a twin in that manner. Of course you could not prove anything that would make the top 20 but you probably could prove n<=10K in a reasonable amount of time and the chances are very small that you would find anything larger than that anyway.

I am checking for 3 (from Riesel primes) and +3 (from Proth primes) twins for the Riesel list of 20101126 and the Proth list of 20101124. I have found 4470 primes or PRPs so far, the largest being 5789*2^15513+3 (4674 digit PRP, and the Primo primality test is queued; of course 5789*2^15513
+1 is a known prime, so this makes a twin PRP/prime pair). None so far are of a reportable size, either for being a top 10000 PRP or being a top 20 twin. I'm sure most of the small ones are already known, just because they're so small you wouldn't have to notice their somewhat special form to run across them, but there are still a good number that are hundreds to thousands of digits that were most likely not previously known. All primes are attached, in two files. The ones in pfgwprime.log were trivially proven to be prime by PFGW, the ones in pfgw.log were found to be PRP. Of the 1773 PRPs there, I have verified the 1755 smallest and submitted the certificates to the FactorDB. (it actually took close to 30 of those and verified the rest itself; it doesn't bother taking outside certificates for primes under 300 digits, but it does enter it into its system for it to verify) The rest are still running in Primo.
The PRP testing is currently at n=475K. There are 275 candidates remaining. In the event (with a dismally low chance of under 3%) that one of those is prime, I'll have found a probable twin prime. AFAIK that would only be officially recognized as a nontwin PRP on the top PRPs list, as it would be too large to prove, but it would still be highly likely to be the largest pair of twin primes known.